Beneficial mutation selection balance and the effect of linkage on positive selection

Abstract

When beneficial mutations are rare, they accumulate by a series of selective sweeps. But when they are common, many beneficial mutations will occur before any can fix, so there will be many different mutant lineages in the population concurrently. In an asexual population, these different mutant lineages interfere and not all can fix simultaneously. In addition, further beneficial mutations can accumulate in mutant lineages while these are still a minority of the population. In this article, we analyze the dynamics of such multiple mutations and the interplay between multiple mutations and interference between clones. These result in substantial variation in fitness accumulating within a single asexual population. The amount of variation is determined by a balance between selection, which destroys variation, and beneficial mutations, which create more. The behavior depends in a subtle way on the population parameters: the population size, the beneficial mutation rate, and the distribution of the fitness increments of the potential beneficial mutations. The mutation-selection balance leads to a continually evolving population with a steady-state fitness variation. This variation increases logarithmically with both population size and mutation rate and sets the rate at which the population accumulates beneficial mutations, which thus also grows only logarithmically with population size and mutation rate. These results imply that mutator phenotypes are less effective in larger asexual populations. They also have consequences for the advantages (or disadvantages) of sex via the Fisher-Muller effect; these are discussed briefly.

Figure 1.—

For beneficial mutations to be acquired by a population, they must both arise and fix. (a) A small asexual population in the successional-mutations (or strong-selection weak-mutation) regime. Mutation A arises early on. Provided it survives drift, it fixes quickly, before another beneficial mutation occurs. Some time later, a second mutation B occurs and fixes. Evolution continues by this sequential fixation process. (b) A larger population in the concurrent-mutations (strong-selection strong-mutation) regime. A mutation A occurs, but before it can fix another mutation B occurs and the two interfere. Here a second mutation, C, occurs in an individual that already has the first mutation A and these two begin fixing together, driving the single mutants to extinction. These dynamics continue with further mutations, such as E and F, occurring in the already-double-mutant population. The key process is how quickly mutations arise in individuals that already have other mutations. This picture has elements of both clonal interference and multiple mutations, illustrated separately in c and d. (c) The clonal interference effect in large populations: a weak-effect beneficial mutation A occurs and begins to sweep, but is outcompeted by a later but more-fit mutation B, which in turn is outcompeted by mutation C. C fixes before any larger mutations can occur; the process can then begin again. Multiple mutations are ignored here. (d) The multiple-mutation effect: several mutations, A, B, and C, of identical effect occur and begin to spread. Mutant lineage B happens to get a second beneficial mutation D, which helps it sweep, outcompeting A and C. Eventually this lineage gets a third beneficial mutation E. Mutations that occur in less-fit lineages, or those that do not happen to get additional mutations soon enough (such as BDF), are driven extinct.

Figure 2.—

Schematic of the evolution of large asexual populations. Shown are fitness distributions within a population, on a logarithmic scale. (a) The population is initially clonal. Beneficial mutations of effect s create a subpopulation at fitness s, which drifts randomly until after time τ₁ it reaches a size of order , after which it behaves deterministically. (b) This subpopulation generates mutations at fitness 2s. Meanwhile, the mean fitness of the population increases, so the initial clone begins to decline. (c) A steady state is established. In the time it takes for new mutations to arise, the less-fit clones die out and the population moves rightward while maintaining an approximately constant lead from peak to nose, qs (here q = 5). The inset shows the leading nose of the population.

Figure 3.—

Schematic of a typical fitness distribution on a logarithmic scale. The total population size is large: . At the front of the distribution—the nose—where only a few individuals are present, stochastic effects are strong but nonlinear saturation is not. The reverse is true in the bulk of the distribution. Stochasticity is strong only when a subpopulation size n is small, , and saturation is strong only when a subpopulation size is large, n ∼ N. Thus there is a wide intermediate regime where neither one matters. We can therefore use a nonlinear deterministic model in the bulk of the distribution, use a linear stochastic model at the front, and match the two in the intermediate regime where both are valid. The bulk of the distribution is dominated by selection, which gives rise to a steady-state Gaussian shape except near the nose.

Figure 4.—

(a) The definition of the establishment time τ_est. A single mutant individual is assumed to exist at t = 0. It drifts stochastically until it either goes extinct or eventually gets large enough that it grows exponentially and its behavior becomes roughly deterministic. We define τ_est to be the inferred time at which the population would have reached size if one extrapolated backward from the long-time deterministic behavior. Note that τ_est is not the time the population actually reached size (indeed, τ_est can be negative). (b) The definition of τ_q: the time between successive establishments of the lead population with fitness qs more than the mean. Mutations occur with a rate that grows exponentially with time. Here, τ_q is the time the new lead population would have reached size , extrapolating backward from its long-time deterministic behavior. This includes both the time to generate a mutant destined to establish and the time for it to drift to substantial frequency.

Figure 5.—

Comparisons between simulations and our theoretical predictions for the mean speed of adaptation v (measured in increase in fitness per generation, × 10⁵). (a) Speed of adaptation v vs. log₁₀[N] for U_b = 10⁻⁵ and s = 0.01. Both the large-N (Equation 41) and the moderate-N (Equation 47) theoretical results are shown in their regimes of validity, which are above and below N ≈ 1/U_b, respectively (the crossover between the two regimes is indicated). (b) v vs. log₁₀[U_b] for N = 10⁶ and s = 0.01. (c) v vs. log₁₀[s] for N = 10⁶ and U_b = 10⁻⁵. Each simulation result shown is the mean v between generations 1500 and 5000 of the simulation, averaged over 30 independent runs. Beginning the average at 1500 generations ensures that in all cases the evolution has reached the steady-state mutation-selection balance (as verified from the time-dependent simulation data, not shown).

Figure 6.—

Comparisons between simulations and our theoretical predictions for the mean q. (a) q vs. log₁₀[N] for U_b = 10⁻⁵ and s = 0.01. (b) q vs. log₁₀[U_b] for N = 10⁶ and s = 0.01. (c) q vs. log₁₀[s] for N = 10⁶ and U_b = 10⁻⁵. All the simulation results are averages of ∼30 independent simulations, after the steady state has been established, as in Figure 5.